Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Log returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of gross returns

vmr vhr vmrl pmr phr mmr mhr vm_ph_r vh_pm_r
Min. : 0.868 0.849 0.801 0.904 0.878 0.988 0.977 0.979 0.967
1st Qu.: 1.044 1.039 1.013 1.042 1.068 1.013 1.013 1.021 1.012
Median : 1.097 1.099 1.085 1.084 1.128 1.085 1.113 1.102 1.094
Mean : 1.070 1.085 1.061 1.065 1.095 1.066 1.087 1.081 1.074
3rd Qu.: 1.136 1.160 1.128 1.107 1.182 1.101 1.128 1.121 1.106
Max. : 1.168 1.214 1.193 1.141 1.208 1.133 1.207 1.178 1.163

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.988 mmr 1.068 phr 1.128 phr 1.095 phr 1.182 phr 1.214 vhr
0.979 vm_ph_r 1.044 vmr 1.113 mhr 1.087 mhr 1.160 vhr 1.208 phr
0.977 mhr 1.042 pmr 1.102 vm_ph_r 1.085 vhr 1.136 vmr 1.207 mhr
0.967 vh_pm_r 1.039 vhr 1.099 vhr 1.081 vm_ph_r 1.128 vmrl 1.193 vmrl
0.904 pmr 1.021 vm_ph_r 1.097 vmr 1.074 vh_pm_r 1.128 mhr 1.178 vm_ph_r
0.878 phr 1.013 vmrl 1.094 vh_pm_r 1.070 vmr 1.121 vm_ph_r 1.168 vmr
0.868 vmr 1.013 mmr 1.085 vmrl 1.066 mmr 1.107 pmr 1.163 vh_pm_r
0.849 vhr 1.013 mhr 1.085 mmr 1.065 pmr 1.106 vh_pm_r 1.141 pmr
0.801 vmrl 1.012 vh_pm_r 1.084 pmr 1.061 vmrl 1.101 mmr 1.133 mmr

Correlations and covariance

Correlations

vmr vhr pmr phr
vmr 1.000 0.993 -0.197 -0.095
vhr 0.993 1.000 -0.119 -0.016
pmr -0.197 -0.119 1.000 0.957
phr -0.095 -0.016 0.957 1.000

Covariances

vmr vhr pmr phr
vmr 0.007 0.009 -0.001 -0.001
vhr 0.009 0.011 -0.001 0.000
pmr -0.001 -0.001 0.004 0.007
phr -0.001 0.000 0.007 0.011

Compare pension plans

Risk of max loss

Risk of max loss of x percent for a single period (year).
x values are row names.

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
0 21.167 21.333 11.833 14.000 12.333 12.667 16.667 16.000
5 12.167 13.167 5.667 8.333 5.833 3.833 8.667 8.167
10 7.000 8.000 3.000 5.000 2.833 0.500 4.333 4.167
25 1.333 1.500 0.500 1.000 0.333 0.000 0.333 0.333
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Worst ranking for loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.333 vhr 13.167 vhr 8.000 vhr 1.500 vhr 0 vmr 0 vmr 0 vmr
21.167 vmr 12.167 vmr 7.000 vmr 1.333 vmr 0 vhr 0 vhr 0 vhr
16.667 vm_ph_r 8.667 vm_ph_r 5.000 phr 1.000 phr 0 pmr 0 pmr 0 pmr
16.000 vh_pm_r 8.333 phr 4.333 vm_ph_r 0.500 pmr 0 phr 0 phr 0 phr
14.000 phr 8.167 vh_pm_r 4.167 vh_pm_r 0.333 mmr 0 mmr 0 mmr 0 mmr
12.667 mhr 5.833 mmr 3.000 pmr 0.333 vm_ph_r 0 mhr 0 mhr 0 mhr
12.333 mmr 5.667 pmr 2.833 mmr 0.333 vh_pm_r 0 vm_ph_r 0 vm_ph_r 0 vm_ph_r
11.833 pmr 3.833 mhr 0.500 mhr 0.000 mhr 0 vh_pm_r 0 vh_pm_r 0 vh_pm_r

Chance of min gains

Chance of min gains of x percent for a single period (year).
x values are row names.

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
0 78.833 78.667 88.167 86.000 87.667 87.333 83.333 84.000
5 63.833 66.667 71.667 76.000 71.667 70.167 69.333 69.000
10 40.833 50.167 32.500 59.667 35.500 46.000 47.167 43.833
25 0.000 0.000 0.000 0.000 0.000 0.833 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Best ranking for gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
88.167 pmr 76.000 phr 59.667 phr 0.833 mhr 0 vmr 0 vmr
87.667 mmr 71.667 pmr 50.167 vhr 0.000 vmr 0 vhr 0 vhr
87.333 mhr 71.667 mmr 47.167 vm_ph_r 0.000 vhr 0 pmr 0 pmr
86.000 phr 70.167 mhr 46.000 mhr 0.000 pmr 0 phr 0 phr
84.000 vh_pm_r 69.333 vm_ph_r 43.833 vh_pm_r 0.000 phr 0 mmr 0 mmr
83.333 vm_ph_r 69.000 vh_pm_r 40.833 vmr 0.000 mmr 0 mhr 0 mhr
78.833 vmr 66.667 vhr 35.500 mmr 0.000 vm_ph_r 0 vm_ph_r 0 vm_ph_r
78.667 vhr 63.833 vmr 32.500 pmr 0.000 vh_pm_r 0 vh_pm_r 0 vh_pm_r

MC risk percentiles

Risk of loss from first to last period.

_m is medium.
_h is high.

a is simulation from estimated distribution of returns of mix.
b is mix of simulations from estimated distribution of returns from individual funds.

l for “long”, going back to 2007.

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
0 4.91 2.65 1.92 0.98 1.18 0 0.63 0.69
5 4.32 2.37 1.68 0.88 1.04 0 0.56 0.55
10 3.67 2.04 1.47 0.80 0.90 0 0.44 0.47
25 2.33 1.16 1.09 0.56 0.55 0 0.22 0.27
50 0.82 0.38 0.63 0.33 0.24 0 0.04 0.13
90 0.05 0.02 0.14 0.07 0.03 0 0.00 0.02
99 0.00 0.00 0.05 0.01 0.00 0 0.00 0.00

1e6 simulation paths of mhr_b:

0 5 10 25 50 90 99
prob_pct 0.118 0.095 0.076 0.036 0.008 0 0

Worst ranking for MC loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.91 vmr 4.32 vmr 3.67 vmr 2.33 vmr 0.82 vmr 0.14 pmr 0.05 pmr
2.65 vhr 2.37 vhr 2.04 vhr 1.16 vhr 0.63 pmr 0.07 phr 0.01 phr
1.92 pmr 1.68 pmr 1.47 pmr 1.09 pmr 0.38 vhr 0.05 vmr 0.00 vmr
1.18 mmr 1.04 mmr 0.90 mmr 0.56 phr 0.33 phr 0.03 mmr 0.00 vhr
0.98 phr 0.88 phr 0.80 phr 0.55 mmr 0.24 mmr 0.02 vhr 0.00 mmr
0.69 vh_pm_r 0.56 vm_ph_r 0.47 vh_pm_r 0.27 vh_pm_r 0.13 vh_pm_r 0.02 vh_pm_r 0.00 mhr
0.63 vm_ph_r 0.55 vh_pm_r 0.44 vm_ph_r 0.22 vm_ph_r 0.04 vm_ph_r 0.00 mhr 0.00 vm_ph_r
0.00 mhr 0.00 mhr 0.00 mhr 0.00 mhr 0.00 mhr 0.00 vm_ph_r 0.00 vh_pm_r

MC gains percentiles

Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
0 95.09 97.35 98.08 99.02 98.82 100.00 99.37 99.31
5 94.49 96.90 97.89 98.85 98.63 100.00 99.29 99.16
10 93.80 96.46 97.75 98.63 98.45 100.00 99.16 98.99
25 91.24 95.08 97.10 98.27 97.63 100.00 98.60 98.41
50 85.83 92.13 95.35 97.17 95.88 99.99 97.39 96.62
100 71.80 83.79 88.74 94.41 89.61 99.59 91.79 90.50
200 39.30 61.73 59.83 85.16 59.41 92.88 69.79 64.10
300 16.30 39.66 23.06 71.21 22.97 71.44 41.76 32.81
400 5.33 23.14 4.29 54.62 3.78 43.88 19.93 12.51
500 1.44 12.68 0.48 38.57 0.25 22.42 7.79 3.76
1000 0.00 0.26 0.02 2.34 0.00 0.23 0.01 0.00

1e6 simulation paths of mhr_b:

0 5 10 25 50 100 200 300 400 500 1000
prob 99.882 99.854 99.824 99.686 99.301 97.513 86.912 65.992 41.486 21.693 0.086

Best ranking for MC gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 mhr 100.00 mhr 100.00 mhr 100.00 mhr 99.99 mhr 99.59 mhr
99.37 vm_ph_r 99.29 vm_ph_r 99.16 vm_ph_r 98.60 vm_ph_r 97.39 vm_ph_r 94.41 phr
99.31 vh_pm_r 99.16 vh_pm_r 98.99 vh_pm_r 98.41 vh_pm_r 97.17 phr 91.79 vm_ph_r
99.02 phr 98.85 phr 98.63 phr 98.27 phr 96.62 vh_pm_r 90.50 vh_pm_r
98.82 mmr 98.63 mmr 98.45 mmr 97.63 mmr 95.88 mmr 89.61 mmr
98.08 pmr 97.89 pmr 97.75 pmr 97.10 pmr 95.35 pmr 88.74 pmr
97.35 vhr 96.90 vhr 96.46 vhr 95.08 vhr 92.13 vhr 83.79 vhr
95.09 vmr 94.49 vmr 93.80 vmr 91.24 vmr 85.83 vmr 71.80 vmr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
92.88 mhr 71.44 mhr 54.62 phr 38.57 phr 2.34 phr
85.16 phr 71.21 phr 43.88 mhr 22.42 mhr 0.26 vhr
69.79 vm_ph_r 41.76 vm_ph_r 23.14 vhr 12.68 vhr 0.23 mhr
64.10 vh_pm_r 39.66 vhr 19.93 vm_ph_r 7.79 vm_ph_r 0.02 pmr
61.73 vhr 32.81 vh_pm_r 12.51 vh_pm_r 3.76 vh_pm_r 0.01 vm_ph_r
59.83 pmr 23.06 pmr 5.33 vmr 1.44 vmr 0.00 vmr
59.41 mmr 22.97 mmr 4.29 pmr 0.48 pmr 0.00 mmr
39.30 vmr 16.30 vmr 3.78 mmr 0.25 mmr 0.00 vh_pm_r

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
m 0.048 0.063 0.058 0.084 0.059 0.082 0.067 0.062
s 0.120 0.126 0.123 0.121 0.088 0.071 0.091 0.090
nu 3.304 4.390 2.265 3.185 2.773 89.863 4.660 3.892
xi 0.034 0.019 0.477 0.018 0.029 0.770 0.048 0.019
R^2 0.993 0.995 0.991 0.964 0.890 0.961 0.927 0.933

Fit statistics ranking

m ranking s ranking R^2 ranking
0.084 phr 0.071 mhr 0.995 vhr
0.082 mhr 0.088 mmr 0.993 vmr
0.067 vm_ph_r 0.090 vh_pm_r 0.991 pmr
0.063 vhr 0.091 vm_ph_r 0.964 phr
0.062 vh_pm_r 0.120 vmr 0.961 mhr
0.059 mmr 0.121 phr 0.933 vh_pm_r
0.058 pmr 0.123 pmr 0.927 vm_ph_r
0.048 vmr 0.126 vhr 0.890 mmr

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000
vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
mc_m 295.32 409.52 345.08 601.31 345.65 541.75 411.22 378.07
mc_s 135.03 210.06 116.43 272.65 108.44 173.87 156.51 137.95
mc_min 4.82 2.88 0.00 1.04 1.45 160.90 34.17 4.87
mc_max 947.71 1593.81 2517.59 2247.55 748.29 1459.68 1230.60 1069.74
dao_pct 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00
dai_pct 4.44 2.39 1.73 0.89 1.03 0.00 0.53 0.62

Ranking

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
601.31 phr 108.44 mmr 160.90 mhr 2517.59 pmr 0.00 vmr 0.00 mhr
541.75 mhr 116.43 pmr 34.17 vm_ph_r 2247.55 phr 0.00 vhr 0.53 vm_ph_r
411.22 vm_ph_r 135.03 vmr 4.87 vh_pm_r 1593.81 vhr 0.00 phr 0.62 vh_pm_r
409.52 vhr 137.95 vh_pm_r 4.82 vmr 1459.68 mhr 0.00 mmr 0.89 phr
378.07 vh_pm_r 156.51 vm_ph_r 2.88 vhr 1230.60 vm_ph_r 0.00 mhr 1.03 mmr
345.65 mmr 173.87 mhr 1.45 mmr 1069.74 vh_pm_r 0.00 vm_ph_r 1.73 pmr
345.08 pmr 210.06 vhr 1.04 phr 947.71 vmr 0.00 vh_pm_r 2.39 vhr
295.32 vmr 272.65 phr 0.00 pmr 748.29 mmr 0.03 pmr 4.44 vmr

Compare Gaussian and skewed t-distribution fits

Gaussian fits

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
m 0.064 0.077 0.061 0.085 0.062 0.081 0.076 0.069
s 0.081 0.099 0.063 0.101 0.048 0.070 0.062 0.060

Gaussian QQ plots

Gaussian vs skewed t

Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
P_norm(X_min) 0.571 0.758 0.511 1.676 5.971 6.842 5.945 4.228
P_norm(X_max) 13.230 11.876 12.922 15.359 9.628 6.429 7.796 8.592
P_t(X_min) 5.377 5.080 3.489 4.315 10.570 8.015 13.008 10.520
P_t(X_max) 0.118 0.156 2.825 0.188 0.488 5.141 0.229 0.175

Average number of years between min or max events (respectively):

vmr vhr pmr phr mmr mhr vm_ph_r vh_pm_r
norm: avg yrs btw min 175.248 131.911 195.568 59.669 16.748 14.616 16.820 23.650
norm: avg yrs btw max 7.559 8.420 7.739 6.511 10.386 15.556 12.827 11.639
t: avg yrs btw min 18.596 19.687 28.663 23.173 9.461 12.476 7.688 9.506
t: avg yrs btw max 848.548 640.410 35.400 531.552 205.104 19.450 437.280 572.483

Velliv medium risk (vmr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.2262221 0.3361598

Objective function plots

Velliv medium risk (vmrl), 2007 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.5098519 0.4248085

Objective function plots

Velliv high risk (vhr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.2965857 0.3073935

Objective function plots

PFA medium risk (pmr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.8351623 0.4382935

Objective function plots

PFA high risk (phr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.2921756 0.3005586

Objective function plots

Mix medium risk (mmr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.7011721 0.3095095

Objective function plots

Mix high risk (mhr), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.4606802 0.3586853

Objective function plots

Mix vmr+phr (vm_ph), 2011 - 2023

QQ Plot

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Parameters

## [1] 1.3815768 0.3410388

Objective function plots

Comments

(Ignoring mhr_a…)

mhr has some nice properties:
- It has a relatively high nu value of 90, which means it is tending more towards exponential tails than polynomial tails. All other funds have nu values close to 3, except phr which is even worse at close to 2. (Note that for a Gaussian, nu is infinite.)
- It has the lowest losing percentage of all simulations, which is better than 1/6 that of phr.
- It has a DAO percentage of 0, which is the same as mmr, and less than phr.
- Only phr has a higher mc_m.
- It has a smaller mc_s than the individual components, vhr and phr.
- It has the highest xi of all fits, suggesting less left skewness. Density plots for vmr, phr and mmr have an extremely sharp drop, as if an upward limiter has been applied, which corresponds to extremely low xi values. The density plot for mhr is by far the most symmetrical of all the fits. As seen in the section “Compare Gaussian and skewed t-distribution fits”, the other skewed t-distribution fits don’t capture the max observed returns at all.
- Only mmr has as higher mc_min. However, that of mmr is 18 times higher with 62, so mmr is a clear winner here.
- Naturally, it has a mc_max smaller than the individual components, vhr and phr, but ca. 1.5 times higher then mmr.
- All the first 4 moments converge nicely. For all other fits, the 4th moment doesn’t seem to converge.

Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.

And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”

Appendix

Many simulations of mc_mhr_b

1e6 paths:

# Down-and-out simulation:
# Probability of down-and-out: 0 percent
# 
# Mean portfolio index value after 20 years: 478.339 kr.
# SD of portfolio index value after 20 years: 163.093 kr.
# Min total portfolio index value after 20 years: 2.233 kr.
# Max total portfolio index value after 20 years: 1561.965 kr.
# 
# Share of paths finishing below 100: 0.1181 percent

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): 0.02798047 
## s(data_x): 0.4736139 
## m(data_y): 10.00934 
## s(data_y): 3.435938 
## 
## m(data_x + data_y): 5.018661 
## s(data_x + data_y): 1.861301

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
100.395 100.523 7.828 8.207
100.280 100.324 7.876 8.423
100.678 100.278 7.797 8.348
100.436 100.097 7.717 8.076
100.251 100.164 7.579 8.266
100.202 100.174 7.870 8.530
100.682 100.875 7.833 8.249
100.401 100.435 7.438 8.372
99.944 100.607 7.766 8.203
100.409 100.310 7.787 8.231 
##       m_a              m_b             s_a             s_b       
##  Min.   : 99.94   Min.   :100.1   Min.   :7.438   Min.   :8.076  
##  1st Qu.:100.26   1st Qu.:100.2   1st Qu.:7.730   1st Qu.:8.213  
##  Median :100.40   Median :100.3   Median :7.792   Median :8.258  
##  Mean   :100.37   Mean   :100.4   Mean   :7.749   Mean   :8.291  
##  3rd Qu.:100.43   3rd Qu.:100.5   3rd Qu.:7.831   3rd Qu.:8.366  
##  Max.   :100.68   Max.   :100.9   Max.   :7.876   Max.   :8.530

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05943   Min.   :0.03653  
##  1st Qu.:0.06521   1st Qu.:0.06151  
##  Median :0.06913   Median :0.06847  
##  Mean   :0.07044   Mean   :0.06724  
##  3rd Qu.:0.07616   3rd Qu.:0.07457  
##  Max.   :0.08687   Max.   :0.09149

The meaning of xi

The fit for mhr has the highest xi value of all. This suggests right-skew:

Max vs sum plot

If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).

If not, \(X\) doesn’t have a \(p\)’th moment.

See Taleb: The Statistical Consequences Of Fat Tails, p. 192